
theorem Th10:
  for L be Abelian left_zeroed add-associative associative
right_zeroed right_complementable distributive non empty doubleLoopStr for a
  being Element of L, p,q being sequence of L holds a*(p*'q)=(a*p)*'q
proof
  let L be Abelian left_zeroed add-associative associative right_zeroed
  right_complementable distributive non empty doubleLoopStr;
  let a being Element of L, p,q being sequence of L;
  for x being Element of NAT holds (a*(p*'q)).x = ((a*p)*'q).x
  proof
    let i be Element of NAT;
    consider f1 be FinSequence of the carrier of L such that
A1: len f1 = i+1 and
A2: (p*'q).i = Sum f1 and
A3: for k be Element of NAT st k in dom f1 holds f1.k = p.(k-'1) * q.(
    i+1-'k) by POLYNOM3:def 9;
    consider f2 be FinSequence of the carrier of L such that
A4: len f2 = i+1 and
A5: ((a*p)*'q).i = Sum f2 and
A6: for k be Element of NAT st k in dom f2 holds f2.k = (a*p).(k-'1) *
    q.(i+1-'k) by POLYNOM3:def 9;
A7: dom (a*f1) = dom f1 by POLYNOM1:def 1
      .= dom f2 by A1,A4,FINSEQ_3:29;
A8: for k be Nat st k in dom f2 holds f2.k = (a*f1).k
    proof
      let k be Nat such that
A9:   k in dom f2;
A10:  k in dom f1 by A1,A4,A9,FINSEQ_3:29;
      then
A11:  p.(k-'1) * q.(i+1-'k) = f1.k by A3
        .= f1/.k by A10,PARTFUN1:def 6;
      thus f2.k = (a*p).(k-'1) * q.(i+1-'k) by A6,A9
        .= a*p.(k-'1) * q.(i+1-'k) by POLYNOM5:def 4
        .= a*(f1/.k) by A11,GROUP_1:def 3
        .= (a*f1)/.k by A10,POLYNOM1:def 1
        .= (a*f1).k by A7,A9,PARTFUN1:def 6;
    end;
    thus (a*(p*'q)).i = a*(Sum f1) by A2,POLYNOM5:def 4
      .= Sum (a* f1) by BINOM:4
      .= ((a*p)*'q).i by A5,A7,A8,FINSEQ_1:13;
  end;
  hence thesis by FUNCT_2:63;
end;
